#6356790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,12)(4,9)(5,10)(6,21)(7,11)(8,15)(13,28)(14,23)(16,29)(17,32)(18,25)(19,27)(20,26)(22,30)(24,31)(33,34,39,44)(35,45,49,55)(36,50,52,57)(37,42,38,43)(40,47,56,48)(41,53,58,51)(46,60,63,54)(59,61,64,62);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,21,29)(4,17,18,31,19,13,20,30)(9,24,26,32,27,22,25,28)(33,35,37,48,39,49,38,47)(34,40,43,45,44,56,42,55)(36,46,51,62,52,63,53,61)(41,54,50,59,58,60,57,64);(33,36,39,52)(34,41,44,58)(35,46,49,63)(37,51,38,53)(40,54,56,60)(42,57,43,50)(45,59,55,64)(47,61,48,62);(1,4,7,19)(2,9,11,27)(3,13,16,17)(5,20,6,18)(8,22,23,24)(10,26,21,25)(12,28,29,32)(14,31,15,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,45,56,55)(41,57,58,50)(46,61,63,62)(54,59,60,64);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,19,20)(8,12,23,29)(9,25,27,26)(13,30,17,31)(22,32,24,28)(33,37,39,38)(34,42,44,43)(35,48,49,47)(36,51,52,53)(40,55,56,45)(41,57,58,50)(46,62,63,61)(54,64,60,59)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263986);